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Why Rubber is incompressible?

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Why rubber and like soft materials are incompressible? I do not want any explanation in formula like, volumetric strain is zero or poissons ratio is 0.5 etc. Physically whats happening when we apply compressive load? For example take a gas, when you compress, the density will change. Is there any of the properties are changing?

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Compressibility is defined as the ability to reduce volume under hudrostatic compression. Or it will come as a function of the bulk modulus. From the strength of materials equations, you can see that when poissons ratio is 0.5, bulk modulus will go to infinity( or to a high value if it is close to 0.5).

A rubber is a network of polymer chains. Each polymer chain consists of many monomers. The polymer chains are crosslinked by covalent bonds. The covalent bonds give the solid-like behavior of the rubber. If these crosslinks are removed, the rubber becomes a polymer melt, and is a liquid.

Thus, a rubber is very similar to a liquid at the level of monomers. Like a liquid, the polymers are densely packed, so that the rubber is difficult to change volume. Also like a liquid, the polymers can move relative to one another, so that the rubber is easy to change shape.

Given this molecular picture, it is clear that the rubber is much easier to change shape than change volume. The shear modulus is much smaller than the bulk modulus. In modeling, we often neglect the change in volume, and focus on change in shape. That is, we assume that the rubber is incompressible.

This idealization of incompressibility is not always suitable. For example, an incompressible material will not support longitudinal wave. But we know rubber can support longitudinal wave. The speed of the longitudinal wave is much larger than the speed of the shear wave.

The specific gravity of rubber is 1.1 to 1.2. That indicates that the ratio of C/N atoms to H atoms in a rubber is approximately equal to the ratio of O atoms to H atoms in water (for a given volume). I find it difficult to reconcile that with the dense packing argument. Any comments?

Dear Biswajit: The phrase "dense packing" is used here in comparison with gas. For example, water is often modeled as being incompressible, too. Water changes shape so readily that its compressibility is only important in specialized situations.

In water molecule, there are two H atoms (atomic weight ~1) for each O atom (wt ~16). In rubber, the mer composition is going to be "crazy," in the sense: "variable." From material to material and perhaps even over the same mer. However, consider that C and N atoms have atomic wts of ~12 and ~14. So, in rubber, I would suppose that the ratio of C/N atoms to H atoms would be much greater than 0.33, for two reasons: (i) a specific gravity > 1.0, and (ii) the lower atomic wts of C/N compared to O. Thus, the two ratios cannot be approximately equal.

As written in a recent post on my personal blog, liquids may be taken to be almost fully densely packed. The volumetric shrinkage during solidification of metals is, say, 10%. Therefore, when a certain quantity of solid metal melts, the extra space available for its atoms to roam around in is, in lineal terms, just about 3.2%. If you take a hard-spheres model, this much is too small an increment for the dense-packing argument not to apply. If a solid material is densely packed, then very nearly so is the same material in its liquid state, too.

(An aside: As a further interesting implication: When flow occurs, there must be granules/cells/"balls," comprising of hundreds of millions or even billions of atoms, all moving together, with individual atoms within a given granule never changing the specific atoms they have as neighbours. Such granules would have to be capable of carrying some kind of a structure.)

Neat explanation, esp. the point regarding dense packing of liquids, and the relation between compressibility and longitudinal waves that you point out. However, I would like to comment on the following:

"Also like a liquid, the polymers can move relative to one another, so that the rubber is easy to change shape."

Liquids have no persistent shape. Solid polymers do. Thus, the respective mechanisms for shape changes have to be different---the former is "permanent," the latter is (more or less) elastic.

The term "relative" motion is somewhat ambiguous. Isn't solid rubber's (or more generally, solid polymeric materials') elastic deformation primarily due to the stretching of the initially kinked polymer chains, as in contrast to a sliding motion of one chain as a whole, against another? Indeed, the very existence of cross-linkages rules out sliding. However, the fact that the cross-linkages occur at isolated points also allows for kinks to be developed in the portions between the linking points, within each mer.

Dear Ajit: Thank you. I should have been more precise. The crosslinks fix the topology of the polymer network. That is, the polymer chains cannot change their connectivity with one another. However, the crosslinks do not prevent small segments of the individual chains from moving relative to one another.

That is, a rubber acts as an elastic solid globally, but acts as a liquid locally.

You're looking for a micromechanical explanation, but from that standpoint rubber is in fact quite compressible. Rubber has a bulk modulus fully two orders of magnitude lower than that of steel, yet steel is treated as compressible.

It's only in considering typical deformations that it becomes useful to consider rubber as incompressible. As Zhigang said, rubber has a high ratio of bulk modulus (K) to shear modulus (G), so its typical deformations are dominated by shear. Assuming it to be totally incompressible usually introduces little error.

Sorry for the equation, but K/G=2(1+v)/(3(1-2v)). So v->0.5 implies large K/G but not large K.

In highly confined geometries (like an O-ring in a groove), compressibility of rubber can become important.

I feel it is pertinent to mention a related difficulty I had, while studying Solid Mechanics and Fluid Mechanics as an undergraduate student. In Solid Mechanics we take the Poisson's ratio of steel to be 0.3, which makes it highly compressible, whereas in Fluid Mechanics, we consider water to be incompressible. This appeared as totally non-intuitive to me, though I didn't have any proper facts and figures with me to approach someone for an explanation.

It remained as an unsolved puzzle, for many years probably, until I recognised that the assumption of compressibility/incompressibility is based on the comparison with other deformations (say shear deformation). The shear strain in steel is of the same order as volumetric strain, whereas shear strain in a flowing liquid is of many orders of magitude higher than the volumetric strain in it.

I do mention this point in my Solid Mechanics class, but I am not sure how well I could convey the matter to my students or how many of them really understand it...